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We present a novel analytic extraction of high-order post-Newtonian (pN)parameters that govern quasi-circular binary systems. Coefficients in the pNexpansion of the energy of a binary system can be found from correspondingcoefficients in an extreme-mass-ratio inspiral (EMRI) computation of the change$\Delta U$ in the redshift factor of a circular orbit at fixed angularvelocity. Remarkably, by computing this essentially gauge-invariant quantity toaccuracy greater than one part in $10^{225}$, and by assuming that a subset ofpN coefficients are rational numbers or products of $\pi$ and a rational, weobtain the exact analytic coefficients. We find the previously unexpectedresult that the post-Newtonian expansion of $\Delta U$ (and of the change$\Delta\Omega$ in the angular velocity at fixed redshift factor) haveconservative terms at half-integral pN order beginning with a 5.5 pN term. Thisimplies the existence of a corresponding 5.5 pN term in the expansion of theenergy of a binary system.  Coefficients in the pN series that do not belong to the subset just describedare obtained to accuracy better than 1 part in $10^{265-23n}$ at $n$th pNorder. We work in a radiation gauge, finding the radiative part of the metricperturbation from the gauge-invariant Weyl scalar $\psi_0$ via a Hertzpotential. We use mode-sum renormalization, and find high-order renormalizationcoefficients by matching a series in $L=\ell+1/2$ to the large-$L$ behavior ofthe expression for $\Delta U$. The non-radiative parts of the perturbed metricassociated with changes in mass and angular momentum are calculated in theSchwarzschild gauge.

Finding high-order analytic post-Newtonian parameters from a high-precision numerical self-force calculation